Approximation by nonfundamental sequences of translates

Approximation by Nonfundamental Sequences of Translates

For functions fit) satisfying certain growth conditions, we consider a sequence of the form {/(c„ — /)}, nonfundamental in L^R), and find a representation for those functions which are in the closure of its linear span. Some theorems concerning degree of approximation are also proved. In [1], we found necessary and sufficient conditions for a sequence of the form {ficn — t)} to be fundamental i...

Approximation by Translates of a Radial Function

Multivariate Interpolation and Approximation by Translates of a Basis Function

This contribution will touch the following topics: Short introduction into the theory of multivariate interpolation and approximation by nitely many (irregular) translates of a (not necessarily radial) basis function, motivated by optimal recovery of functions from discrete samples. Native spaces of functions associated to conditionally positive definite functions, and relations between such sp...

Student Seminar Talk APPROXIMATION BY TRANSLATES OF A FUNCTION

We discuss the following problem: given φ ∈ L2(R) and a sequence Λ ⊂ R, is it possible that the system of translates {φ(x − λ)}λ∈Λ spans the whole space, and how sparse the spectrum Λ might be? 1. Systems of translates 1.1. Let a function φ ∈ L(R) and a discrete sequence Λ ⊂ R be given. We are interested to know if the system of translates {φ(x− λ)}λ∈Λ (1.1) is complete in L(R), that is, if the...

Nonlinear Approximation by Sums of Exponentials and Translates

In this paper, we discuss the numerical solution of two nonlinear approximation problems. Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem. Let h be a linear combination of exponentials with real frequencies. Determine all frequencies, all coefficients, and the number of summands if finitely many perturbed, uniformly sampled ...